1. Field of the Invention
The present invention generally relates to a wavelets-based multiresolution representation of a 3D image object, and more particularly, to a wavelets-based multiresolution representation of a 3D image object, which is capable of compressing and storing the 3D image object, and gradually transmitting and effectively representing the 3D image, by rearranging 3D meshes in an irregular arrangement for an application of wavelets method.
2. Description of the Prior Art
Explosion of Internet use, stunning advancement of network field and introduction of 3D image equipments have brought 3D image objects in the spotlight of the industrial field. The problem of using such 3D image object is that it consumes a considerable amount of capacity. The 3D image object also causes many loads for transmission. Further, displaying the 3D image object in display devices is performed at a quite slow speed, and thus it consumes a lengthy time. A wavelets-based multiresolution representation of 3D image object can solve the above problems.
In order to express 3D image object in 3D space at multiresolutions by using wavelet scheme, a basis function has to be defined first. The basis function is expressed with a scaling function Φj(x) and a wavelet Ψj(x) as follows:                                           Φ            j                    ⁡                      (            x            )                          =                  [                                                                      ϕ                  0                  j                                ⁡                                  (                  x                  )                                            ,                                                           ⁢              …                        ⁢                                                   ,                                          ϕ                                                      m                    j                                    -                  1                                j                            ⁡                              (                x                )                                              ]                                    [                  Mathematical          ⁢                                           ⁢          expression          ⁢                                           ⁢          1                ]            
where, φij(x) has a value ‘1’ in the coordinate i of the 3D mesh at a resolution (j), and has a value ‘0’ in the other coordinates. The wavelet is expressed by             Ψ      i      j        ⁡          (      x      )        =            ϕ                        2          ⁢          i                +        1                    j        +        1              ,  and determined depending on the scaling function Φj(x). In such functions, there are two matrices according to the multiresolution principle, satisfying the following relation:                               [                                    Φ                              j                -                1                                      |                          Ψ                              j                -                1                                              ]                =                              Φ            j                    ⁡                      [                                          P                j                            |                              Q                j                                      ]                                              [                  Mathematical          ⁢                                           ⁢          expression          ⁢                                           ⁢          2                ]            
Also, by the filter bank algorithm, the following equation is obtained.                                           [                                                            Φ                                      j                    -                    1                                                  ⁡                                  (                  x                  )                                            |                                                Ψ                                      j                    -                    1                                                  ⁡                                  (                  x                  )                                                      ]                    ⁡                      [                                          A                j                                            B                j                                      ]                          =                              Φ            j                    ⁡                      (            x            )                                              [                  Mathematical          ⁢                                           ⁢          expression          ⁢                                           ⁢          3                ]            
Based on the mathematical expressions 2 and 3, relation of the analysis filters Aj, Bj and synthesis filter Pj, Qj is expressed by,                                           [                                          P                j                            |                              Q                j                                      ]                                -            1                          =                  [                                    A              j                                      B              j                                ]                                    [                  Mathematical          ⁢                                           ⁢          expression          ⁢                                           ⁢          4                ]            
In order to prove orthogonal relation between the basis functions and construct wavelets in 3D space, inner products of the functions have to be newly defined. Lounsbery functions f and g have inner products that can be expressed by the following equation:                                           <                    ⁢          f          ,                                           ⁢          g          ⁢                      >                          =                              ∑                          t              ∈                              Δ                ⁡                                  (                                      M                    0                                    )                                                                                                     ⁢                                           ⁢                                    1                              Area                ⁡                                  (                  τ                  )                                                      ⁢                                          ∫                                  X                  ∈                  τ                                                                                               ⁢                                                f                  ⁡                                      (                    x                    )                                                  ⁢                                  g                  ⁡                                      (                    x                    )                                                  ⁢                                                                   ⁢                                  ⅆ                  x                                                                                        [                  Mathematical          ⁢                                           ⁢          expression          ⁢                                           ⁢          5                ]            
where, Δ(M0) is a group of triangles constituting original meshes M, and τ indicates a signal triangle constituting M0. This definition starts from the fact that all the triangles of the 3D meshes have equal areas. The following interrelation can be derived from the mathematical expressions 2 and 5.                               I          j                =                                            (                              P                                  j                  +                  1                                            )                        T                    ⁢                      I                          j              +              1                                ⁢                      P                          j              +              1                                                          [                  Mathematical          ⁢                                           ⁢          expression          ⁢                                           ⁢          6                ]            
where, Ij is an inner product between two basis functions at a resolution j.
Surface subdivision of 3D image object surface can be performed with the application of wavelets scheme. While subdividing 3D meshes in 1:4 ratio using the functions defined for multiresolution representation in 3D space, filters that are indicated by matrices are generated. FIG. 1 is a view showing such interrelation. Referring to FIG. 1, each triangle is split into 4 divisions sequentially. Exponents of value M 0, 1, 2 indicate times of split, while the exponents of value Q 1, 2 indicate synthesis filters from each of the split stages.
Initially, the 3D image meshes are arranged in irregular coordinates. For example, one coordinate is connected to four triangles, lines and coordinates, while another coordinate is connected to the two triangle, three lines and coordinates. These meshes cannot be represented at wavelets-based multiresolution representation, unless they are rearranged in a regular pattern. The ‘remeshing’ algorithm enables the use of wavelet scheme by rearranging the coordinates of the meshes of such irregular arrangement.
By the ‘remeshing’ algorithm based on a mapping, 3D image data are converted into an image in 2D space, reconstructed into a new image of certain regularity that can be applied with the wavelet scheme by parameterization, and then retransformed into an image of 3D space. Such image enables the multiresolution image representation.
Eck proposed remeshing algorithm based on harmonic mapping. The harmonic mapping provides mathematical tools that enable the conversion into 2D image with a maximum distortion of the 3D image surface.
When there is b: Db→Pb that converts bounds of the 3D image patches into a 2D convex image, h: D→P, a harmonic map that satisfies the function b is found. A solution can be obtained by minimizing the following energy function of the harmonic map:                               E          ⁡                      (            h            )                          =                              1            2                    ⁢                                    ∑                                                                                 ⁢                                                      (                                          i                      ,                                                                                           ⁢                      j                                        )                                    ∈                                      Edges                    ⁡                                          (                      D                      )                                                                                                                                         ⁢                                                   ⁢                                          K                ij                            ⁢                                                                                                            h                      ⁡                                              (                                                  v                          i                                                )                                                              -                                          h                      ⁡                                              (                                                  v                          j                                                )                                                                                                              2                                                                        [                  Mathematical          ⁢                                           ⁢          expression          ⁢                                           ⁢          7                ]            
where, h(vi), h(vj) are coordinates of original 3D image (D), vi,vj are coordinates of 2D image obtained through harmonic map, and Kij is a spring constant disposed along the edge of the image D.
However, the methods proposed by Lounsbery or Eck are based on the condition that all the triangle constituting the 3D image meshes have the equal areas. Accordingly, Lounsbery or Eck cannot effectively express the areas that are geometrically complex or have severe convex. Further, the Eck's remeshing takes considerable time to obtain the patches of given meshes. It is also problematic as several steps of parameterization need to be taken.